Red-Black Trees. 2/24/2006 Red-Black Trees 1

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1 Red-Black Trees 3 8 //00 Red-Black Trees 1

2 Outline and Reading From (,) trees to red-black trees ( 10.5) Red-black tree ( ) Definition Height Insertion restructuring recoloring Deletion restructuring recoloring adjustment //00 Red-Black Trees

3 From (,) to Red-Black Trees A red-black tree is a representation of a (,) tree by means of a binary tree whose nodes are colored red or black In comparison with the (,) tree, a red-black tree has same logarithmic time performance simpler implementation with single node type (rather than -,3- & - nodes) OR 5 7 //00 Red-Black Trees 3

4 Red-Black Tree A red-black tree can also be defined as a binary search tree that satisfies the following properties: Root Property: the root is black External Property: eery leaf is black Internal Property: the children of red nodes are black Depth Property: all the leaes hae the same black depth //00 Red-Black Trees

5 Height of a Red-Black Tree Theorem: A red-black tree storing n items has height O(log n) Proof: The height of a red-black tree is at most twice the height of its associated (,) tree, which is O(log n) The search algorithm for a red-black tree is the same as that for a binary search tree By the aboe theorem, searching in a red-black tree takes O(log n) time //00 Red-Black Trees 5

6 Insertion To perform operation insert(k, o), we execute the insertion algorithm for binary search trees and color red the newly inserted node, unless it is the root We presere the root, external, and depth properties If the parent of is black, we also presere the internal property and we are done Else ( is red ) we hae a double red (i.e., a iolation of the internal property), which requires a reorganiation of the tree Example where the insertion of causes a double red: //00 Red-Black Trees

7 Remedying a Double Red Consider a double red with child and parent, and let w be the sibling of Case 1: w is black The double red is an incorrect replacement of a -node Restructuring: we change the -node replacement Case : w is red The double red corresponds to an oerflow Recoloring: we perform the equialent of a split w 7 w //00 Red-Black Trees 7

8 Restructuring A restructuring remedies a child-parent double red when the parent red node has a black sibling It is equialent to restoring the correct replacement of a -node The internal property is restored and the other properties are presered w 7 7 w //00 Red-Black Trees 8

9 Restructuring (cont.) There are four restructuring configurations depending on whether the double red nodes are left or right children //00 Red-Black Trees 9

10 Recoloring A recoloring remedies a child-parent double red when the parent red node has a red sibling The parent and its sibling w become black and the grandparent u becomes red, unless it is the root It is equialent to performing a split on a 5-node The double red iolation may propagate to the grandparent u w u 7 w u //00 Red-Black Trees 10

11 Analysis of Insertion Algorithm insert(k, o) 1. We search for key k to locate the insertion node. We add the new item (k, o) at node and color red 3. while doublered() if isblack(sibling(parent())) restructure() return else { sibling(parent() is red } recolor() Recall that a red-black tree has O(log n) height Step 1 takes O(log n) time because we isit O(log n) nodes Step takes O(1) time Step 3 takes O(log n) time because we perform O(log n) recolorings, each taking O(1) time, and at most one restructuring taking O(1) time Thus, an insertion in a redblack tree takes O(log n) time //00 Red-Black Trees 11

12 Deletion To perform operation remoe(k), we first execute the deletion algorithm for binary search trees Let be the internal node remoed, w the external node remoed, and r the sibling of w If either of r was red, we color r black and we are done Else ( and r were both black) we color r double black, which is a iolation of the internal property requiring a reorganiation of the tree Example where the deletion of 8 causes a double black: 3 8 r w 3 r //00 Red-Black Trees 1

13 Remedying a Double Black The algorithm for remedying a double black node w with sibling y considers three cases Case 1: y is black and has a red child We perform a restructuring, equialent to a transfer, and we are done Case : y is black and its children are both black We perform a recoloring, equialent to a fusion, which may propagate up the double black iolation Case 3: y is red We perform an adjustment, equialent to choosing a different representation of a 3-node, after which either Case 1 or Case applies Deletion in a red-black tree takes O(log n) time //00 Red-Black Trees 13

14 Red-Black Tree Reorganiation Insertion Red-black tree action restructuring recoloring Deletion Red-black tree action restructuring recoloring adjustment remedy double red (,) tree action change of -node representation split remedy double black (,) tree action transfer fusion change of 3-node representation result double red remoed double red remoed or propagated up result double black remoed double black remoed or propagated up restructuring or recoloring follows //00 Red-Black Trees 1